Spin-lattice relaxation via quantum tunneling in diluted crystals of Fe
4single-molecule magnets
A. Repoll´es,1,2A. Cornia,3,*and F. Luis1,2,†
1Instituto de Ciencia de Materiales de Arag´on, CSIC-Universidad de Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain 2Departamento de F´ısica de la Materia Condensada, Universidad de Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain 3Dipartimento di Scienze Chimiche e Geologiche and UdR INSTM, Universit`a di Modena e Reggio Emilia, via G. Campi 183,
41125 Modena, Italy
(Received 3 December 2013; published 26 February 2014)
We investigate the dynamic susceptibility of Fe4 single-molecule magnets with integer spin (S= 5) in the
form of pure crystals as well as diluted in crystals of isostructural, but nonmagnetic, Ga4 clusters. Below
approximately 1 K, the spin-lattice relaxation becomes dominated by a temperature-independent process. The spin-lattice relaxation time τ measured in this “quantum regime” is 12 orders of magnitude shorter than the characteristic time scale of direct phonon-induced processes but agrees with the relaxation times of pure (i.e., not assisted by phonons) spin tunneling events. The present results show that the latter phenomenon, despite conserving the energy of the ensemble of electronic and nuclear spins, drives the thermalization of electronic spins at very low temperatures. The spin-lattice relaxation time scales with the concentration of Fe4, thus suggesting
that the main effect of dipolar interactions is to block tunneling. The data show therefore no evidence for the contribution of collective phonon emission processes, such as phonon superradiance, to the spin-lattice relaxation. DOI:10.1103/PhysRevB.89.054429 PACS number(s): 75.45.+j, 76.30.Kg, 75.50.Xx, 75.40.Gb
Single molecule magnets (SMMs) [1] are high-spin mag-netic molecules comprising one or more metal centers encap-sulated in a shell of organic ligands. They provide a very attractive workbench for research on quantum phenomena in magnetism, such as magnetization tunneling [2–4], Berry phase interferences [5], quantum spin coherence [6–8], and quantum phase transitions [9]. Although the underlying physics governing such phenomena is fairly well understood, some fundamental questions still remain open.
A particularly intriguing puzzle concerns the nature of spin-lattice relaxation (SLR) mechanisms that bring spins to thermal equilibrium at very low temperatures, typically for T 1 K, when thermally activated processes [10–12] die out. Under these conditions and near zero magnetic field, spins predominantly flip by tunneling across the anisotropy energy barrier. Hyperfine interactions with environmental nuclear spins (e.g., those of the metal centers themselves and of other atoms present in the outer ligand shell) can compensate for the magnetic bias associated with intercluster dipolar interactions, thus bringing some molecular spins close to resonance conditions and enabling them to tunnel [13–15]. Tunneling modifies the magnetization but conserves the energy of the ensemble of nuclear and electronic spins. By contrast, SLR requires that magnetic energy is either released to or absorbed from the lattice, e.g., via the direct emission or absorption of a phonon [16]. Since the latter events can be extremely slow at low magnetic fields, it can be expected that magnetization dynamics and SLR take place at very different time scales [17].
Yet, experiments performed on different SMMs [18–20] give SLR times that are close to the expected tunneling times, thus suggesting that the thermalization of electronic spins is dictated by tunneling fluctuations. A plausible, yet qualitative interpretation of the existing experimental evidences is that
*acornia@unimore.it †fluis@unizar.es
SLR takes place via phonon superradiance [21] from partic-ular spin configurations, which the spin ensemble “visits” via tunneling processes [22]. This phenomenon has been investigated on lanthanide ions diluted in diamagnetic crystals [22,23]. Unfortunately, the results are obscured by either the dependence of the quantum tunnel splitting on concentration (for Kramer’s ions) or the existence of large hyperfine splittings, which dominate the physics at very low T . These effects prevent any simple, quantitative comparison of SLR experiments with theoretical predictions for spin tunneling.
Crystals of polynuclear SMMs can provide a valuable alternative. However, synthesizing crystalline solid solutions of intact polynuclear species and their diamagnetic analogues is a very challenging task. First, preparing isostructural but diamagnetic variants of known SMMs may be difficult or impossible, especially for mixed-valent species. Second, the solid solution must crystallize without metal scrambling, i.e., without any exchange of metals that produces mixed-metal species. The first successful synthesis of diluted polynuclear SMMs in crystalline form was achieved [24] with tetra-iron molecular clusters (see Fig.1), which are known to be highly stable and robust [25–27]. A fraction of Fe4 clusters (S= 5)
is replaced with nonmagnetic but structurally equivalent Ga4 clusters, thereby providing a means of controlling the
characteristic energy scale of dipolar couplings. In addition, the vast majority (98%) of Fe atoms carry no nuclear spin. These crystals are therefore model systems to study the relationship between SLR and tunneling, and to elucidate the role played by dipolar interactions. In this paper, we study the SLR times of pure as well as diluted Fe4 crystals and
compare the results with existing theories for SLR and spin tunneling.
Single-crystalline samples of [(Fe4)x(Ga4)1−x(L)2(dpm)6]·
C6H6, hereafter referred to as (Fe4)x(Ga4)1−x with x= 1.00
and 0.05, were prepared as described in Ref. [24]. Com-pounds of the series (Fe4)x(Ga4)1−x are isomorphous and
crystallize in the monoclinic space group C2/c with four equivalent molecules per unit cell. Only minor variations
(a)
(c)
(b)
FIG. 1. (Color online) (a) Molecular structure of Fe4 viewed
along the normal to the metal plane, which is taken to coincide with the easy magnetic axis (Z). (b) Magnetic energy level scheme of Fe4(dotted horizontal lines). Levels are labeled by the m quantum
number associated with the Z component of the spin in the absence of tunneling effects. The solid line is the classical double-well potential as a function of m. (c) Crystal structure of Fe4 SMMs diluted in a
diamagnetic crystal of Ga4 clusters. Color code: Fe, light orange;
Ga, dark blue; O, red; C, gray. Hydrogen atoms and benzene lattice molecules have been omitted for clarity. The normal to the metal plane (Z) forms an angle of 1.44◦with a∗.
of unit cell parameters are observed in the series, with a 1% contraction of unit cell volume from x= 1.00 to x = 0 at 120 K. Magnetic measurements in the region of 1.8 K T 300 K were performed on powder specimens using a commercial SQUID magnetometer. Ac susceptibility measurements have been extended down to 13 mK using a μ-SQUID susceptometer installed inside the mixing chamber of a 3He-4He dilution refrigerator [28,29]. In these experi-ments,∼800 × 400 × 200 μm3single crystals were directly placed on top of one of the two μ-SQUID loops. The easy magnetic axis made an angle ψ 56◦ with respect to the ac excitation magnetic field (hac<1 mOe). The frequency of the
latter was varied between ω/2π = 0.01 Hz and 2 × 105Hz.
Representative examples of the in-phase χ and out-of-phase χ magnetic susceptibilities of Fe4 and
(Fe4)0.05(Ga4)0.95, measured as a function of frequency at
fixed temperatures, are shown in Fig. 2. They show the typical behavior of a SMM, with a well defined transition from equilibrium conditions, at sufficiently low frequencies, to adiabatic conditions, in the opposite frequency limit. The
FIG. 2. (Color online) Frequency dependent susceptibility isotherms of pure Fe4 (left) and of (Fe4)0.05(Ga4)0.95 (right) at
several temperatures. Solid lines are least-square Cole-Cole fits [cf. Eqs. (1) and (2)]. The insets show the temperature dependence of the reciprocal in-phase susceptibility jump 1/χ . Solid lines are least-square Curie-Weiss fits, with the parameters shown in each graph.
transition takes place approximately when ωτ 1, where τ is the SLR relaxation time, and coincides with the maximum of χ. We find (see the insets of Fig. 2) that the in-phase susceptibility “jump” χ (i.e., the net variation between its high and low frequency limits) follows Curie-Weiss law χ C/(T − θ), where C is the Curie constant and θ is the Weiss temperature that depends on the average strength of intermolecular magnetic interactions. This shows that linear susceptibility experiments measure SLR to thermal equi-librium and not spin-spin relaxation within the “spin-bath.” This experimental situation contrasts sharply with that met in spin tunneling experiments, be it magnetization hysteresis or Landau-Zener relaxation measurements. In the latter case, magnetization jumps that occur at tunneling resonances (level crossings) link two nonequilibrium spin configurations. As expected, C scales with x since it is proportional to the number of spins per unit of sample mass. In addition, θ of Fe4is about
six times larger than that of (Fe4)0.05(Ga4)0.95, thus showing
that interactions become significantly reduced by dilution. Above T 1 K, SLR times were obtained by fitting susceptibility isotherms with Cole-Cole functions [30]
χ= χS+(χT − χS) 1+ (ωτ)βcos (βπ/2) 1+ 2 (ωτ)βcos (βπ/2)+ (ωτ)2β (1) χ= (χT − χS) (ωτ ) β sin (βπ/2) 1+ 2 (ωτ)βcos (βπ/2)+ (ωτ)2β, (2) where χT and χSare the equilibrium and adiabatic suscepti-bilities, respectively, and β gives information on the width of the distribution of relaxation times. We find that β ranges from 0.85 at T = 2.1 K to 0.79 at T = 1.2 K for pure Fe4and from
0.82 at T = 2.1 K to 0.72 at T = 1.2 K for (Fe4)0.05(Ga4)0.95.
For temperatures below 1 K, the maxima of χ occur at frequencies lower than our minimum experimental limit
(0.01 Hz). Even so, SLR times can be approximately estimated using the relation τ [r/(sin βπ/2 − r cos βπ/2)](1/β)/ω, which follows from Eqs. (1) and (2). Here, r = χ/(χ− χS), and β and χShave been fixed to their respective values found at T = 1.2 K, as they are expected to vary only weakly with T . The results are shown in Fig. 3. SLR relaxation times of both samples follow a thermally activated behavior above approximately 1.2 K, where τ τ0exp(U/kBT), gradually
approaching a nearly constant value below 1 K. However, some differences are seen between the two samples. The activation energy U is largest for the pure Fe4crystal, which also has the
shortest prefactor τ0. The same trend was found by in-field
(0.1 T) ac measurements on powder samples [24]. In the temperature-independent, or quantum, regime, the relaxation of the pure Fe4 compound is about 10 times slower than that
of the diluted sample.
We next compare these results with predictions for the characteristic time scales of phonon-induced relaxation and pure tunneling processes. Let us first consider the spin Hamiltonian of an isolated Fe4cluster. Magnetic interactions
between the four Fe3+ions within the cluster core, shown in Fig.1, give rise to a ground state with a net spin S= 5 and a gyromagnetic ratio g= 2.005(4). Interactions with the crystal field and with magnetic fields split this multiplet. These effects can be described by the following spin Hamiltonian:
H = B0
2O20+ B22O22+ B40O40− gμB−→S ·−→H + H, (3)
where B20/kB= 0.200(2) K, B22/kB= 0.023(1) K, and
B0
4/kB= 9(4) × 10−6K are magnetic anisotropy parameters,
FIG. 3. (Color online) Spin-lattice relaxation times of pure Fe4
(top) and (Fe4)0.05(Ga4)0.95(bottom). Solid symbols were obtained
from Cole-Cole fits [Eqs. (1) and (2)] of isothermal susceptibility vs frequency measurements. Open symbols were obtained from the ratio
χ/(χ− χS), as described in the text. Solid lines are phonon-induced
relaxation times calculated by solving a Pauli master equation for the population of magnetic energy levels. The dotted lines are spin tunneling times predicted by Eq. (4).
determined from the fit of spectroscopic data, and H is an effective term arising from interactions that mix states from different multiplets [24]. The ensuing energy level scheme is shown in Fig.1. The population of the ground state level doublet, associated with the maximum projections m= ±S of the spin along the anisotropy axis Z, becomes larger than 99% for T 1 K. Under these conditions, each Fe4cluster behaves
effectively as a two level system, with an energy splitting E= (2+ ξ2)1/2, where is the quantum tunnel splitting induced by off-diagonal terms in Eq. (3) and ξ 2gμBSHzis
the energy bias associated with longitudinal magnetic fields. In Ref. [24], it was reported that /kB 9 × 10−7K and that
it depends only weakly on transverse magnetic fields. In order to realistically account for the influence of intermolecular magnetic interactions, we have also evaluated the distributions of internal dipolar fields in concentrated and diluted samples. Two model crystal samples were tested: (i) a spherical portion of crystal lattice with radius 300 ˚A and (ii) a model crystal mimicking the typical crystal shape bound by faces (001), (111), and (111) (and their equivalents) at distances of 20, 16, and 8 interplanar spacings, respectively, from the origin. The two models comprised comparable num-bers of complete molecules (∼4.3 − 4.5 × 104). To describe
magnetically-diluted samples, a fraction 1− x of molecules was randomly chosen and removed from the ensemble. Molecular spins were randomly assigned to be either in the m= +5 or in the m = −5 states, with equal probability. The dipolar field −H→d= (Hd,X,Hd,Y,Hd,Z) acting on each
molecule was computed from the dipolar fields experienced by its constituent ions. The longitudinal component Hd,Z
was evaluated as the weighted average over the four ions, whereas for the transverse (XY ) component an unweighted average was used [24,31]. Owing to the zero magnetization, the demagnetizing field vanishes, and calculations made on the spherical and nonspherical model samples give virtually identical results (see Fig.4). For the pure Fe4sample, the bias
field distribution is Gaussian with FWHM of approximately 20 mT, while for the diluted sample it is non-Gaussian and much narrower (FWHM∼ 2 mT). The transverse fields distribution for x= 1 is rather broad and extends up to 30 mT with a peak at ca. 10 mT. By contrast, the diluted sample features a more structured distribution extending up to about 10 mT, with a main peak at 0.5 mT and secondary maxima at higher field values. These arise from pairs of neighboring Fe4
complexes in the lattice.
SLR times have been numerically computed by applying a Pauli master equation to calculate the time-dependent populations of the magnetic energy levels of Eq. (3) and, from them, the frequency-dependent susceptibility [10,32]. The spin-phonon interaction Hamiltonian was 3B20{( xz+
ωxz)⊗ [SxSz+ SzSx]+( yz+ ωyz)⊗ [SySz+ SzSy]}, where iz and ωiz represent phonon-induced strains and rotations, respectively. The overall scale of all relaxation rates is fixed by a constant parameter q∝ nph/c5s, where nphis the number of
relevant low-energy phonon modes and csis the speed of sound
that describes, in the simplest manner, the dispersion relation ω= cskof phonon modes. This parameter was determined by
fitting τ measured at T = 2 K on pure Fe4. The speed of sound
(a)
(b)
FIG. 4. (Color online) (a) Shaded areas, distribution of dipolar bias fields Hd,Z calculated for magnetically unpolarized crystals
of Fe4 (light red) and (Fe4)0.05(Ga4)0.95 (dark blue); lines,
least-squares fits of these distributions using either a Gaussian (x= 1) or a Lorentzian (x= 0.05) function. (b) Distributions of transverse dipolar magnetic fields calculated for the same model samples.
nph = 3 acoustic modes) to 1.5 × 105cm/s (if all 768 acoustic
and optical modes contribute). These values are comparable to those previously found for other SMMs [18,20,32]. In order to “tune” the ground state tunnel splitting to its actual value while keeping the numerical calculations tractable, we simulated the effect of H by introducing an off-diagonal B65O65 term into the giant spin Hamiltonian of the S= 5 multiplet, with B65/kB= 1.05 × 10−6K. The susceptibility has been averaged
over the bias field distribution P (Hd,Z) shown in Fig.4. For
simplicity, transverse dipolar interactions have been taken into account by introducing an average transverse magnetic field Hd,⊥(1/√2,1/√2,0), with Hd,⊥= 10 mT and 0.5 mT for the
concentrated and diluted samples, respectively.
The results of these calculations are shown in Fig.3. Above 1 K, they account well for the experimental data: At any temperature, thermally activated relaxation becomes faster as the magnetic concentration x decreases. In particular, the effective activation energies (obtained from fits for T > 1 K) are U/kB 12.9 K for x = 1 and U/kB 12 K for x =
0.05. Both values are smaller than the “classical” activation energy Ucl/kB= 15.0 K extracted from spectroscopic data
through Eq. (3) [24], thus showing that tunneling via thermally activated spin states strongly influences τ in this regime. By contrast, the same model completely fails to account for the SLR observed below 1 K. It predicts a monotonic increase of τ with decreasing temperature down to approximately 0.1 K, where it tends to saturate to an astronomically long (1013 s) value, associated with direct phonon-induced
processes. Clearly, a different process drives SLR below 1 K. At very low temperatures, spin dynamics is fully dominated by pure quantum spin tunneling events. According to the Prokof’ev and Stamp model [13], the average tunneling rate is approximately given by
=
2
P(ξd), (4)
where P (ξd)= P (Hd,Z)/(2gμBS) is the distribution of dipolar
energy bias. Tunneling times −1 obtained from Eq. (4) for pure and diluted samples agree very well with the SLR times measured below 1 K. In particular, at zero applied field, Eq. (4) predicts (dotted lines in Fig. 3) that −1 approximately scales with the width of P (ξd), i.e., with x,
as the SLR time indeed does. This result contrasts with the decrease of τ with increasing Er3+ concentration observed in crystals of Na9[ErxY1−x(W5O18)2]·yH2O [22]. Yet, the
markedly different behavior of these materials can also be reconciled with the above interpretation, on the basis of Eq. (4). While of Fe4 is approximately independent of
Hd,⊥, of Er3+, a Kramer’s ion with J = 15/2, vanishes unless the local Hd,⊥ = 0, thus it is expected to increase with
concentration. Also, the slight temperature dependence of τ in the quantum regime can be associated with gradual changes in the distribution of dipolar bias.
The thermalization of spins plays a crucial role in funda-mental phenomena, such as the attainment of magnetically ordered states, as well as in their application as magnetic refrigerants or thermometers, thus its relevance can hardly be overestimated. Our experiments show that spin-lattice relax-ation of anisotropic spins takes place, at very low temperatures, at rates that quantitatively match predictions for spin tunneling processes, thus much faster than those of direct spin-phonon processes. However, the precise mechanism by which the spins that flip by tunneling exchange energy with the lattice remains obscure. For a fixed tunneling rate, such as that of Fe4, dipolar
interactions mainly slow down the relaxation process by taking most spins off-resonance. Therefore, experiments provide no evidence supporting the contribution of collective emission of phonons to spin-lattice relaxation. The last piece of the puzzle thus remains to be found.
This paper has been partly funded by the Spanish MINECO (Grant No. MAT2009-13977-C03), the Gobierno de Arag´on (project MOLCHIP), the European Union (ERANET project NanoSci-ERA: Nanoscience in European Research Area SMMTRANS), the Italian MIUR (PRIN2008 project) and the Universit´e Joseph Fourier (Grenoble, France) through a visiting professorship to A.C.
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